On the concave one-dimensional random assignment problem and Young integration theory

TL;DR

This paper studies the one-dimensional random assignment problem with concave cost functions, establishing the existence of a limit for the normalized costs using Young integration theory, and extends results to the bipartite TSP.

Contribution

It introduces a novel approach using Young integration theory to analyze the assignment problem with concave costs and proves the existence of cost limits for exponents different from 1/2.

Findings

Limit of normalized costs exists for p ≠ 1/2

Young integration theory provides a new analytical tool

Results extend to bipartite Traveling Salesperson Problem

Abstract

We investigate the one-dimensional random assignment problem in the concave case, i.e., the assignment cost is a concave power function, with exponent $0<p<1$, of the distance between $n$ source and $n$ target points, that are i.i.d. random variables with a common law on an interval. We prove that the limit of a suitable renormalization of the costs exists if the exponent $p$ is different than $1/2$. Our proof in the case $1/2<p<1$ makes use of a novel version of the Kantorovich optimal transport problem based on Young integration theory, where the difference between two measures is replaced by the weak derivative of a function with finite $q$-variation, which may be of independent interest. We also prove a similar result for the random bipartite Traveling Salesperson Problem.